Coupled Flexural-Torsional Nonlinear Vibrations of Piezoelectrically Actuated Microcantilevers With Application to Friction Force Microscopy

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1 Couped Fexura-Torsiona Noninear Vibrations of Piezoeectricay Actuated Microcantievers With Appication to Friction Force Microscopy S. Nima Mahmoodi Post Doctorate Nader Jaii 1 Associate Professor e-mai: jaii@cemson.edu Smart Structure and Nanoeectromechanica Systems Laboratory, Department of Mechanica Engineering, Cemson University, Cemson, SC The probem of vibrations of microcantievers has recenty received considerabe attention due to its appication in severa nanotechnoogica instruments, such as atomic force microscopy, nanomechanica cantiever sensors, and friction force microscopy. Aong this ine, this paper undertakes the probem of couped fexura-torsiona noninear vibrations of a piezoeectricay actuated microcantiever beam as a typica configuration utiized in these appications. The actuation and sensing are both faciitated through bonding a piezoeectric ayer (here, ZnO) on the microcantiever surface. The beam is considered to have simutaneous fexura, torsiona, and ongitudina vibrations. The piezoeectric properties combined with noninear geometry of the beam introduce both inear and noninear coupings between fexura vibration as we as ongitudina and torsiona vibrations. Of particuar interest is the inextensibiity configuration, for which the governing equations reduce to couped fexura-torsiona noninear equations with piezoeectric noninearity appearing in quadratic form whie inertia and stiffness noninearities as cubic. An experimenta setup consisting of a commercia piezoeectric microcantiever instaed on the stand of an utramodern aser-based microsystem anayzer is designed and utiized to verify the theoretica deveopments. Both inear and noninear simuation resuts are compared to the experimenta resuts and it is observed that noninear modeing response matches the experimenta findings very cosey. More specificay, the softening phenomenon in fundamenta fexura frequency, which is due to noninearity of the system, is anayticay and experimentay verified. It is aso discosed that the initia twisting in the microcantiever can infuence the vaue of the fexura vibration resonance. The experimenta resuts from a macroscae beam are utiized to demonstrate such twistfexure couping. This unique couping effect may ead to the possibiity of indirect measurement of sma torsiona vibration without the need for any anguar dispacement sensor. This observation coud significanty extend the appication of friction force microscopy to measure the friction of a surface indirecty. DOI: / Keywords: noninear vibration of microcantiever beams, couped fexura-torsiona vibrations, piezoeectricay-actuated microcantievers, friction force microscopy 1 Corresponding author. Contributed by the Technica Committee on Vibration and Sound of ASME for pubication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 1, 006; fina manuscript received January 8, 008; pubished onine October 14, 008. Assoc. Editor: Christopher D. Rahn. 1 Introduction and Probem Statement The probem of vibrations of microcantievers has received a considerabe attention since it appears in severa scientific and industria appications such as atomic force microscopy AFM, nanomechanica cantiever sensors NMCS, and friction force microscopy FFM. Utiizing microcantievers provides the abiity to rapidy detect sma quantities of materias in different environment such as gas or iquid. The microcantiever beam vibration spectrum has been used to detect a target mass partice attached eccentricay to the beam, which produces both fexura and torsiona vibrations 1. In both FFM and AFM, a microcantiever beam with a sharp tip is utiized, as shown in Fig. 1a, for acquiring topographic information from the surface through mechanica interaction between microcantiever tip and sampe,3. Whie mosty fexura vibrations occur in AFM, FFM may experience torsiona vibration in addition to bending. This is due to the fact that the scanning direction in FFM is perpendicuar to beam ength and the tip of the beam. The resuting friction force twists the tip of the beam producing torsion in the beam, as shown in Fig. 1b. The couped fexura-torsion vibrations in beam may mosty occur due to the foowing three conditions: i shear center of the beam being offset from the neutra axis, ii gyroscopic effect, and iii geometry of the beam. The presence of an offset between center of gravity and shear center can ead to a couping between fexura and torsiona vibrations in mechanica structures 4. Research resuts show that for such beam systems, there is a significant effect in natura frequencies, mode shapes, and response due to the fexure-torsion couping 5. This effect appears in inear form, i.e., the couping of fexura-torsiona vibrations eads to inear couped equations of motion. A gyroscopic effect caused by rotation of beam base can aso produce a couped fexuratorsiona vibration in the system 6. In this case, the couping is due to the anguar veocity at the base of the beam. This paper focuses on the fexura-torsion couping due to the geometry and the noninearity appearing in the system. In addition, the presence Journa of Vibration and Acoustics Copyright 008 by ASME DECEMBER 008, Vo. 130 / Downoaded From: on 05/05/015 Terms of Use:

2 Scan direction FFM tip Friction Sampe surface a b Scan direction Twist ange Friction Fig. 1 a Schematic operation of FFM and b twist of the FFM tip of a piezoeectric ayer on the beam for the purpose of actuation and sensing introduces new noninear terms in the equations of motion of the microcantiever beam. Considering the higher order terms of vibration in motion of microcantievers wi resut aso in couping between torsiona and fexura motions. These higher order terms are due to the geometry of the system, which can be attributed due to either argeampitude vibration of the microcantiever or considering the inextensibiity of the beam. By considering the inextensibiity condition, the effect of ongitudina vibration can be imposed into fexura vibration, which brings added noninearity into equations of motion 7. The reason that such geometrica noninearity is considered here is because of the sma scae nature of microcantievers, i.e., they may vibrate with arge ampitude in response to a sma appied force 8. The governing differentia equations of motion of nonpanar, noninear dynamics of an inextensibe beam containing noninearities up to order three have been derived and studied 9. The directy excited osciations of the beam ignoring the torsion effect have aso been anayzed using a perturbation technique 10. In addition, simiar work has been done for noninear vibration of extensibe beam 11. The mathematica mode of the noninear fexura-fexura-torsiona motions of an extensiona Euer Bernoui beam has been presented 1. Negecting the torsiona inertia, the two fexura modes of noninear nonpanar vibrations of inextensibe beams, which are parametricay excited, have been investigated 13. Using Timoshenko beam theory, the fexura vibrations of a microbeam have been investigated 14. The probem of noninear fexura vibration of inextensibe beam and its soution methods have aso been studied in different works Recenty, some nove investigations have been done on couped torsiona-fexura vibration of beams. Using the dynamic transfer matrix method, the natura frequencies and mode shapes of the fexura-torsiona couped vibrations of axiay oaded thinwaed beams with monosymmetrica cross sections have been deveoped 0. The couped fexura-torsiona free and forced vibrations of a beam with tip and/or in-span attachments have been investigated 1. In most of these works, the effect of torsion has been ignored. The piezoeectric patch has been widey utiized as a sensor/ actuator in both macro- and microcantiever beams. The dynamics of cantiever beam actuated with a piezoeectric ayer considering the eectromechanica stress-strain reations have been investigated,3. Experimenta investigation has been done on noninear fexura behavior of aminated piezoeectric actuators as a function of the ampitude and frequency of arge eectric fieds 4. However, the expicit noninear terms due to the presence of piezoeectric in equations of motion need further deveopment and research. Different from these common practices, the noninear couped fexura-torsiona vibrations of microcantiever beam actuated by a piezoeectric patch attached on surface of the beam are investigated in this paper. Numerica simuations and experimenta testing are performed to study the interaction between fexura and torsiona noninear modes and natura frequencies. Using the obtained formuations, a nove method for an indirect measurement of friction force in FFM is introduced and extensivey anayzed. Dynamic System Modeing A uniform and initiay straight metaic microcantiever with a ayer of piezoeectric materia on top of its surface, as depicted in Fig., is considered. The beam foows the Euer Bernoui beam theory, where shear deformation is negigibe. The beam has ength with mass per unit ength, which is cantievered on one end and free at the other end. The x,y,z axes in Fig. are considered to be inertia, whie the,, axes are assumed to be principa coordinates of the beam cross section at the arbitrary position s. Figure 3 depicts the coordinate systems for the microcantiever in which us,t and vs,t are components of dispacement vector s aong the axes x and y, respectivey, and t is the time. The reationship between principa axes and the inertia axes is described by two Euer ange rotations. As depicted in Figs. and 3, s,t and s,t are rotation anges to take x and z to and, respectivey. Three variabes u, v, and are introduced to measure ongitudina, fexura, and torsiona vibrations, respectivey. The fexura ange between x and axes is defined as. Using Fig. 3, for an eement of ength ds can be expressed as = tan u where prime indicates the derivative with respect to position s. By etting overdots denoting the partia derivative with respect to Fig. 3 Fig. Schematic of the microcantiever beam Coordinate systems of the microcantiever beam / Vo. 130, DECEMBER 008 Transactions of the ASME Downoaded From: on 05/05/015 Terms of Use:

3 Fig. 5 Geometry of the microcantiever beam Fig. 4 time t, and utiizing Tayor series expansion, the anguar veocity,, of the beam can be obtained as 9,13 = e + sine + cose = e + u u e +1 1 e where e i is a unit vector and i=x,y,z,,, indicates the direction of the unit vector. The overdot and prime denote a partia derivative with respect to time and position, respectivey. Simiary, the curvature vector of the beam can be written as = e + sine + cose = e + u u e +1 1 e 3 Considering the noninear vibration of the beam, for a point on the neutra axis, the position of the point may be expressed as r 0 =se x. Then, after the deformation, the position of the point moves to r = s + ue + e 4 The strain associated with the materia ocated at neutra axis is given by 0 = Straight and deformed positions of an arbitrary point p r s. r 1/ s r 0 s. r 1/ 0 = 1+u s It is considered here that the beam is inextensibe, which demands no reative eongation of the neutra axis. Therefore, utiizing Eq. 5, it yieds 5 1+u + =1 6 Using Eq. 6 and the Tayor series expansion, it can be concuded that u = 1 The beam is assumed to possess uniform cross-sectiona area. If it is considered that the cross section of the beam is at an arbitrary position s, as shown in Fig. 4, and p is a point on the cross section ocated at, reative to the neutra axis, then after deformation, point p with the dispacement components u, v, and w moves to p*. The coordinates of p* are and because of the assumption that the shape of the cross section remains uniform after deformation. The position vectors for both p and p* can be written as r p = se x + e y + e z 7 r p* = s + ue x + e y + e + e 8 Using the dispacement vectors and definition of strain tensor ij and engineering shear strains ij, one can write 11 =, 1 = 1 =, 9 13 = 13 =, = 3 = 33 =0 Considering the pain strain in the beam and the fact that the ratio of beam width to thickness is very high for such microscae size beam configuration, the moduus of easticity, such as pates, must be corrected in the form of 6 E* E = 1 10 where E* is the moduus of easticity and is Poisson s ratio. As shown in Fig. 5, the piezoeectric ayer is not bounded on the entire ength of the beam; therefore, the neutra surface varies for each section. For s 1 or s where piezoeectric is not attached, the neutra surface is the geometric center of the beam y n =0. The position of neutra surface, y n, for the section where piezoeectric ayer is attached can be obtained as y n = E ph p h p + h b 11 E p h p + E b h b where h indicates the beam thickness and the subscripts b and p denote the beam materia and piezoeectric ayer properties, respectivey, and w is the beam width. It is known that the eectrica and the mechanica phenomena are couped for piezoeectric materias. The stress-strain reations for the piezoeectric materias are expressed as 5 = S E + d T D 1 Q = d + D 13 where S E is the 66 compiance matrix, is the 61 stress vector, d is the 36 matrix of the piezoeectric constants, D is the 31 eectrica dispacement vector, Q is the 31 eectrica fied vector, and is the 33 matrix of the dieectric permitivities. For the microcantiever considered here, the eectrica dispacement is one dimensiona defined as D 1 = D 3 =0, D t 14 Hence, the couping reation between the stress and the eectrica fieds for piezoeectric ayer is expressed as 7 p 11 = E p p Pt 11 E p d h p where Pt is the appied votage to the piezoeectric materia. 3 Governing Equations of Motion An energy method is utiized here to derive the equations of motion. Using the assumptions and preiminary derivations obtained in the previous section, the tota kinetic energy of the system can be presented as Journa of Vibration and Acoustics DECEMBER 008, Vo. 130 / Downoaded From: on 05/05/015 Terms of Use:

4 where T = 1 msu + + J + J 0 + J u u ds 16 ms = w b b h b + H 1 H p h p J s = 1 1 msw b + h b + H 1 H h p J s = 1 1 msw b 17 and C s = H 0 H 1 E b I b + H 1 H E b I b + bt b y n + H 1 H E p I p b + H H E b I I b = w b h b 3 k b, C c s = H 1 H w p E pd 31 h p + h b y n k b = h b 5 w b n=1,3,... I b = w 3 bh b 1 tanh 1 nw b n 5 h b and J s = 1 1 msh b + H 1 H h p H 1 = Hs 1, H = Hs 18 Hs is the Heaviside function, is the inear mass density, and J i is the mass moment of inertia with respect to axis i=,,. The tota potentia energy of the beam and piezoeectric ayer can now be written as U = /A b 11 b 11 + b 1 b 1 + b 13 b 13 dads I p = w p h p 3 k p, I b = w bh b 3 1 k p = h p 5 w p n=1,3,... I p = w 3 ph p 1 tanh 1 nw p n 5 h p I p = w ph p y n h p + h b h p y n + 3h 1 3 p + 3 h bh p h bh p /A b 11 b 11 + b 1 b 1 + b 13 b 13 dads Utiizing the Lagrangian expression presented in Eq. 1, the equations of motion of the system with respect to three variabes u,, and v can be obtained as /A /A p 11 p 11 dads b 11 b 11 + b 1 b 1 + b 13 b 13 dads + 1 EAsu + uv v4ds 19 0 where EAs = H 0 H 1 E b w b h b + H 1 H E p w p h p 0 Using Eqs. 3 16, the Lagrangian of the system can be expressed as L = 1 msu + v + J + J v + J v v u 0 where v vv u v v C s C s v C sv v v v v u vvu + C c sv vu vu vv 1 vpt EAsu + uv v4ds C s = H 0 H 1 G b I b + H 1 H G b I b + bt b y n + H 1 H G p I p b + H H G b I C s = H 0 H 1 E b I b + H 1 H E b I b + H 1 H E p I p + H H E b I b 1 J J v C C v + C 1 C cvpt = J 4a =0 ats =0, =0 ats = 4b v 1 C cvpt + J v + EAu + 1 C v +C vv 1 C cvpt = mü 4c u =0 ats =0, u =0 ats = 4d J u v +vv C uv +vv + EAuv + 1 v3 + C c 1 u + vvpt C v + C v v vv vu vu 1 C c1 u v Pt + t J v + J v v uv u v v v = mv 4e v = v =0 ats =0, v = v =0 ats = 4f As seen from Eq. 4, there exist noninear terms in the equations from order two to three. However, the noninearities of the second order are due to the presence of piezoeectric ayer. Considering Eqs. 4a and 4e, it is observed that the torsiona and fexura vibrations are couped in two ways; one is a third order noninear couping due to beam geometry and the other one is a second order noninearity due to both geometry and eectromechanica / Vo. 130, DECEMBER 008 Transactions of the ASME Downoaded From: on 05/05/015 Terms of Use:

5 couping of piezoeectric ayer. The former noninear terms have been obtained in previous studies, whie the atter is a new observation, which is discosed for the first time here. Having the equations of motion, a number of case studies are considered, next, to study the couping of fexura and torsiona vibrations in the presence of piezoeectric actuator patch and effect of noninearity due to the beam geometry. 3.1 Fexura-Torsiona Vibration. If the ongitudina vibration is ignored, then the equations of motion reduce to the foowing form: J J v C C v + C 1 C cvpt = J 5a =0 ats =0, =0 ats = 5b J vv C vv + 1 EAv3 + C c vvpt + t J v + J v v v v C v + C v v vv 1 C c1 v Pt = mv 5c v = v =0 ats =0, v = v =0 ats = 5d As seen, there sti exists the same couping between bending and torsion in the system. A coser ook at Eqs. 4a, 4b, 5a, and 5b, reveas that there is no effect of ongitudina vibrations in the equation of the torsiona vibrations. This shows that even considering higher order terms in the geometry does not resut in couping between the torsion and ongitudina vibrations. However, the fexura and ongitudina vibrations are couped and ignoring ongitudina vibrations eads to omission of couped noninear terms in Eq. 4e, which eads to Eq. 5c. 3. Fuy Symmetric Uniform Beam. If the beam is considered to be competey symmetric in the sense that J =J and C =C this corresponds, for exampe, to square or circe crosssectiona areas, then the noninear terms in Eq. 4a are removed and torsion vibration is not ony inear but aso decouped from fexura vibration. This concudes that even for arge deformations, the noninear terms in torsiona vibrations wi not appear if the beam cross section is square or circe shape. However, even for such beams, the presence of piezoeectric ayer sti coupes the fexura and torsiona vibrations. In this case, the equations of motion in Eqs. 4a 4f reduce to the foowing form: C 1 C cvpt = J 6a =0 ats =0, =0 ats = 6b v 1 C cvpt + J v + EAu + 1 C v +C vv 1 C cvpt = mü 6c u =0 ats =0, u =0 ats = 6d J u v +vv C uv +vv + EAuv + 1 v3 + C c 1 u + vvpt C v vv vu vu 1 C c1 u v Pt + t J v uv u v v v = mv 6e v = v =0 ats =0, v = v =0 ats = 6f Athough there sti exist some noninear terms in the equations but the ony term, which coupes the fexure and torsion, is the second order noninear term in Eq. 6a and the one in Eq. 6e, which are due to the presence of piezoeectric actuator ayer. Equation 6a possesses no onger the third order noninearity when compared to Eq. 4a. 3.3 Inextensibe Beam. The inextensibiity condition expresses that the eongation of the neutra axis during the vibration is ignorabe. Considering the inextensibiity condition and using Eq. 6, Equation 4c and 4e can be reduced into one equation consequenty. Then, the equations of motion of the system can be presented in the foowing form J J v C C v + C 1 C cvpt = J 7a =0 ats =0, =0 ats= 7b 1 vc cvpt vc vv mv s 0 s v v + v dsds C v + C v v 1 C c1 v Pt + t J v + J v v = mv 8a v = v =0 ats =0, v = v =0 ats = 8b Using the couping of stretching and fexure to the inextensibiity condition, the two noninear equations for ongitudina and fexure vibrations are combined into one. There are now two orders of noninearities: The second order noninearity is due to the presence of piezoeectric ayer and the third order noninear terms are due to the geometry and appear as noninear inertia and stiffness terms. 4 Assumed Mode Mode Expansion In this section, the numerica simuations of the fexuratorsiona noninear vibrations of inextensibe beam are presented. The equations of motion have been presented in Sec Athough two other cases have been introduced in Secs. 3.1 and 3., they are not numericay soved here. The inextensibe beam equations given in Sec. 3.3 represent a genera case in which a three fexura, ongitudina, and torsiona vibrations are investigated a together. For the sake of simpicity, the effect of rotary inertia J,J terms is ignored in this section. Therefore, the equations of motion of system 7a, 7b, 8a, and 8b can be simpified to J C + C C v + 1 C cvpt =0 9a =0 ats =0, =0 ats= 9b Journa of Vibration and Acoustics DECEMBER 008, Vo. 130 / Downoaded From: on 05/05/015 Terms of Use:

6 mv + C v + v s m0 s v v + v dsds + vc vv + C C v + 1 C c v + vc c vpt = 1 C c Pt 9c v = v =0 ats =0, v = v =0 ats = 9d In order to produce the ordinary differentia equations governing the time functions of equations of motion, these equations are separated into position and time components using Gaerkin approximation as s,t = m=1 m s,t = m=1 m sq m t 30 vs,t = v n s,t = n=1 n sr n t 31 n=1 where m and n are the comparison functions satisfying ony beam geometrica boundary conditions and not necessariy the equations of motion 9a 9d for bending and torsion of the microcantiever beam and q n are the generaized time-dependent coordinates. Since the boundary conditions of the beam are camped free, the inear mode shapes for torsion and fexure are considered here to be the foowing comparison functions: m s = A m sin m s cos n cosh n =0 34b The constants A n and B n can be obtained using the orthogonaity conditions. Defining the inner product of two arbitrary functions fx and gx as fx,gx fxgxdx 35 =0 Substituting Eqs. 30 and 31 into Eqs. 9a 9d and taking the inner product using Eq. 35 of the resuting equations with n s and n s, respectivey, yieds where k 1mn q m + k mn q m + k 3mn q m r n + k 4mn q m r n Pt =0 36a k 5mn r n + k 6mn r n + k 7mn r n Pt + k 8mn r 3 n + k 9mn r n r n + r n ṙ n + k 10mn r n q m + k 11mn q m Pt = k 1mn Pt 36b k mn =0 k 3mn =0 k 1mn =0 J s m sds C s m s m sds C s C s m s n sds 37a 37b 37c where n s = B ncosh n s cos n s + sin n s sinh n s cosh n + cos n sin n + sinh n 33 k 4mn = 1 C c s m s n sds 0 k 5mn =0 ms n sds 37d 37e m = m 1 and n are the roots of the frequency equation, 34a k 6mn =0 n sc s n sds 37f Fig. 6 a The Poytec MSA-400 testing device and b the tip section of the microcantiever beam / Vo. 130, DECEMBER 008 Transactions of the ASME Downoaded From: on 05/05/015 Terms of Use:

7 Tabe 1 Physica properties of the microcantiever Symbo Vaue E b 185 GPa 500 m w b 50 m w t 55 m b 330 kg/m 3 t b 4 m b 0.8 E p 133 GPa 375 m w p 130 m d pc/n p 6390 kg/m t p 4 m p 0.5 k 7mn = 1 n sc c s n sds 40 k 8mn =0 k 9mn =0 k 10mn =0 1 n s n C c s n sds 0 n s n sc s n s n sds s s n s n s n sdsds ms0 ds n sc s C s n s m ds 37g 37h 37i 37j a Band 1: 55.5KHz Band 4: 39.5KHz Start : 47.1KHz Start : 35.3KHz End: 65 KHz End: 45.3KHz Width: 17.9 KHz Width: 10 KHz First resonance khz Frequency Response Second resonance 39.7 khz b 50 v(,t) (Micrometer) Frequency (KHz) Fig. 7 a Experimenta resut and b ogarithmic simuation resuts for 1 V chirp excitation signa with first fexura natura frequency highighted Journa of Vibration and Acoustics DECEMBER 008, Vo. 130 / Downoaded From: on 05/05/015 Terms of Use:

8 k 11mn = 1 n sc c s m sds 40 37k k 1mn = 1 n scsds c 0 37 Using Eqs. 37a 37, the noninear couped equations 36a and 36b can be simuated with MATLAB /SIMULINK and the resuts are presented and compared with experiments as discussed next. 5 Experimenta Setup and Methods In this section, an experimenta setup is designed for the noninear couped fexura-torsiona vibrations of the microcantiever beam. The piezoeectric actuator ony excites the fexura mode of vibration, but not the torsion. However, there is an interaction between the fexura and torsiona vibration modes, as demonstrated through extensive modeing in Sec. 3 and hence piezoeectric harmonic force has the potentia to excite the torsion mode aso. For this purpose, a DMASP microcantiever beam made by Veeco Instruments is utiized, as depicted in Fig. 6. The DMASP microcantiever has a smaer width at the end when compared with the rest of the microcantiever see Fig. 6b. There is a sma tip vertica to the eongation of the microcantiever at its end, which is used in AFM appications. The piezoeectric ayer consists of a 3.5 m ZnO ayer and two Ti/Au ayers of 0.5 m height on top and beneath the ZnO ayer at the base of the probe. The physica properties are isted in Tabe 1. In order to perform the experiment, a state-of-the-art microsystem anayzer, the MSA-400 manufactured by Poytec Inc. 8 Fig. 6a, is used for noncontact measurement of threedimensiona motions, especiay the out-of-pane vibration of the microcantiever tip. The microcantiever beam vibration is measured in response to a 1 V ac votage appied to the piezoeectric ayer as the excitation source. 6 Numerica Simuations and Experimenta Resuts This section is divided into two parts. The numerica simuations for the mode and the experimenta resuts are first compared to verify the modeing efforts presented here. More specificay, the softening phenomenon in fundamenta fexura frequency, which is due to noninearity of the system, is demonstrated anayticay. In the second part, the couping between the fexura and torsiona vibrations is numericay studied at the microscae and resuts are discussed. The experimenta resuts from a macroscae beam are used to demonstrate the fexura-torsiona couping effect. These experiments were performed at this scae due to the difficuty in onine measurement of twist at microscae even with our state-of-the-art microsystem anayzer. On the other hand, since the micro- and macrobeams woud have simiar characteristics incuding the fexura-torsiona couping effect, this is an acceptabe exercise and can pace our nove concept for FFM on a firm foundation. 6.1 Preiminary Resuts and Mode Verification. In order to compare the numerica resuts with experiments, the coefficients introduced in Eqs. 9a 9d are cacuated for different mode shapes of fexure and torsion. Having these vaues, the noninear equations 7a and 8b can be numericay simuated. Some of the vaues for k imn have very arge quantities due to the microscae Fig. 8 Simuation resuts for the first three torsiona natura frequencies for 1 V chirp excitation signa nature of beam with its sma dimensions. This can behave highy noninear when arge deformation is appied to the microcantiever due to the actuation of piezoeectric ayer. This actuation depends primariy on the appied votage to piezoeectric materia. Hence, even sma votage can move the vibration of the microscae beam into noninear regime. In order to compare the experimenta and numerica resuts, the beam is experimentay actuated. The resuts for first fexura natura frequency depicted in Fig. 7a show that the first natura frequency of the beam is 55,561 Hz. In addition, the time response is iustrated with 1 V actuation appied to the beam through the piezoeectric ayer. The properties in Tabe 1 have been utiized to perform a numerica simuation to find the frequency response. As seen, the obtained resuts match the experimenta resuts very cosey. The ogarithmic numerica resuts for the first fexura mode are depicted in Figs. 7b, respectivey. It shoud be noted that experimenta resuts are for the tip veocity, whie the numerica ones are for the tip dispacement responses. The focus of comparison here is ony on frequencies and not the ampitude at this stage. The numerica simuations for frequency response of the system with the second fexura natura frequency highighted as forth band is depicted in Fig. 7b. There are two sma peaks in experimenta resuts Fig. 7a. These peaks are due to the subharmonic resonance in the system. They actuay exist in numerica resuts but they are very sma compared to resonance frequencies and aso their ampitude changes due to initia conditions. Figure 8 aso shows the first three noninear natura frequencies of the torsiona vibrations of the microcantiever beam obtained by simuation in the presence of noninear couped geometry and piezoeectric terms. 6. Fexura-Torsiona Couping Effect Indirect Twist Measurement 6..1 Numerica Simuations at Microscae. Equations 36a and 36b demonstrate the couping between the fexura and torsiona vibrations. Using this couping effect, the sma torsiona Tabe Vaues of fexura frequency in response to different initia twist vaues Initia twist anges deg Fexura freq. Hz 55,561 55,575 55,611 55,68 55,789 55,933 56,111 56,374 56,803 57,08 57, / Vo. 130, DECEMBER 008 Transactions of the ASME Downoaded From: on 05/05/015 Terms of Use:

9 Fig. 9 Frequency response simuation of fexura vibration couped with torsiona vibration for different initia twists ange of the beam tip can be measured by investigating the changes in frequency response of the fexura vibration. This nove idea originates from the appication of FFM, as the microcantiever moves aong the scanning direction, as shown in Fig. 1. When the motion becomes stabe, the twisting ange of the beam tip due to friction is considered as initia vaue of torsiona vibration of the beam. This vaue, in turn, changes the torsiona vibration and due to the couping effect, the frequency response of fexura vibration varies. This phenomenon can be utiized to indirecty measure the friction coefficient of the surface through variations of fexura frequency response. In order to demonstrate this fact, the fexura vibration frequency of microcantiever due to the sma initia torsiona anges is provided in Tabe. The vaues in Tabe show that the frequency response of the fexura vibrations nonineary changes due to the increase in initia twists. These changes, even for sma amounts of one-tenth of degree, are easiy measurabe. The frequency response shifts of the fexura vibrations are shown in Fig. 9 for five different initia twist anges seected from Tabe. It is observed that the initia twist causes a shift in fexura frequency response; however, there is a decrease in ampitude of the response due to the increase in initia twist of the microcantiever. In addition, this decreasing curve is noninear in nature and it is demonstrated that for arger amounts it reduces faster. Therefore, the measurement becomes ess sensitive as the initia twist increases. 6.. Experimenta Verification at Macroscae. In this section, the couping of fexura-torsiona vibration for indirect measurement of twist is experimentay studied. As mentioned earier, the experiment is performed on macroscae beams due to the hardware imitations at microscae. The experimenta setup consists of a cantiever beam with a piezoeectric ayer bonded on its top surface acting as the actuation mechanism, and a noncontact aser for tip defection measurement. First, a cantiever beam made of auminum is used for the preiminary experimenta investigation with ength of 0 cm, width of 1 cm, and thickness of 0.4 mm. The piezoeectric ayer is an ACX-QPB, which is attached on Fig. 10 beam the beam using epoxy,.5 cm from the camped end. The vibration of the tip is measured and monitored by a Microepsion optoncdt-1607 aser distance sensor. To cause permanent twisting in the cantiever beam, the beam is twisted unti it goes under pastic deformation. Heat cannot be utiized here due to the presence of piezoeectric patch on the beam. Since for the cantiever beam shoud be twisted to pastic region, the obtained twists are thus random and cannot be equay spaced. After giving a permanent twist deformation, the twist ange is measured using a eve and a goniometer, as shown in Fig. 10. In the experiment with 0 cm ong cantiever beam, since the fundamenta natura frequency is reativey ow around 4.34 Hz, the changes in the natura frequency due to torsion are very sma see Tabe 3. However, even these sma changes in the natura frequency are experimentay observabe due to the initia twists. In order to obtain better resuts, two other cantiever beams with same properties but with engths of 17.5 cm and 15 cm utiized see Tabe Observations. The obtained experimenta resuts show that the initia twist changes the fexura natura frequency. Athough the change is very sma for macroscae cantiever beams due to their sma natura frequencies, it is detectabe experimentay. As numericay demonstrated for microscae beams, the change in natura frequency was more evident due to the arge vaues of natura frequencies. Since at microscae measurement of twist is a ot more difficut in practice, the proposed idea of indirect measurement of twist is very attractive in practice. In addition, it can be observed that the increase in fexura natura frequency due to the initia twist is not inear. see Tabe 3. 7 Concusions Measurement setup for the initia twist in cantiever The noninear equations of motion of couped fexura-torsiona vibrations of a nonhomogenous noninear microcantiever beam have been derived. The noninear terms due to the presence of piezoeectric materia and the couping between the fexura and torsiona vibrations as a resut of noninearity in geometry and the presence of piezoeectric ayer have been identified. The cubic and quadratic noninearities were observed due to the geometry of the Tabe 3 Fundamenta fexura natura frequency variation with different initia twists for beams with different engths Beam ength cm Twist ange deg Natura frequency Hz Journa of Vibration and Acoustics DECEMBER 008, Vo. 130 / Downoaded From: on 05/05/015 Terms of Use:

10 beam and piezoeectric ayer. First, two fexura resonances have been numericay and experimentay obtained and compared. The experimenta resuts matched the numerica ones very cosey with ony a sma error. Moreover, the couping effect has been used to measure the vaue of torsion. It has been demonstrated that the initia vaue of the torsion can change the vaue of the fexura resonance. This property is very hepfu in appication of FFM to indirecty measure the friction of a surface without the use of a torsiona dispacement sensor. Acknowedgment The materias presented here are based on work supported by the Nationa Science Foundation CAREER Grant No. CMMI , and MRI Grant No. CMMI The authors woud aso ike to thank Associate Editor Professor Christopher Rahn and the reviewers for their constructive comments and carefu review that have improved the quaity of the paper significanty. References 1 Sharos, L. 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